×

Hereditary rings containing an injective maximal left ideal. (English) Zbl 0803.16010

R. Yue Chi Ming has proposed the following questions in his earlier papers. (i) If a ring \(R\) contains an injective maximal left ideal and if every maximal left ideal of \(R\) is projective, then is \(R\) semisimple Artinian? (ii) If \(R\) is a left hereditary ring containing an injective maximal left ideal, is \(R\) semisimple Artinian?
The authors answer both the questions in the negative by constructing a hereditary ring \(R\) containing an injective maximal left ideal as well as an injective maximal right ideal such that \(R\) is Artinian with nonzero Jacobson radical \(J\) with \(J^ 2 = 0\). They also give two new characterizations of semisimple Artinian rings in terms of left hereditary rings containing an injective maximal left ideal.

MSC:

16E60 Semihereditary and hereditary rings, free ideal rings, Sylvester rings, etc.
16D25 Ideals in associative algebras
16P20 Artinian rings and modules (associative rings and algebras)
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Yue Chi Ming R., Arch Math
[2] Yue Chi Ming R., Comm. in Algebra 20 (3) pp 749– (1992) · Zbl 0754.16007
[3] Yue Chi Ming R., Riv. Mat. Univ., Parma 8 (3) pp 443– (1982)
[4] Yue Chi Ming R., Ricerche di Matematica (2) pp 147– (1984)
[5] Osofsky B.L., J. Math 14 (2) pp 645– (1964)
[6] Faith C., Algebra I: Rings, Modules and Categories (1981) · Zbl 0508.16001
[7] Rege M.B., Math. Japonica 31 (6) pp 927– (1986)
[8] Anderson F.W., Rings and Categories of Modules (1974) · Zbl 0301.16001
[9] Googearl K.R., Rings Theory: Nonsingular Rings and Modules (1976)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.